Chapters 17 &19 Temperature, Thermal Expansion and The Ideal Gas Law


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1 Chapters 17 &19 Temperature, Thermal Expansion and The Ideal Gas Law Units of Chapter 17 & 19 Temperature and the Zeroth Law of Thermodynamics Temperature Scales Thermal Expansion Heat and Mechanical Work Specific Heats Latent Heats
2 173 Temperature, heat, Thermal Equilibrium and the Zeroth Law of Thermodynamics Definition of Temperature: Temperature is a measure of how hot or cold something is. A hot oven is said to be at HIGH temperature and ice of a frozen lake is said to be at LOW temperature A more correct definition: Temperature is a measure of the average kinetic energy of all particles in an object. Definition of heat: Heat is the energy transferred between objects because of a temperature difference. Objects are in thermal contact if heat can flow between them. E.g. If a hot object is brought into thermal contact with a cold object, heat will be exchanged or thermal energy will transfer from one object to another. When the transfer of heat between objects in thermal contact ceases, they are in thermal equilibrium and the objects are at the same Temperature
3 173 Temperature, heat, Thermal Equilibrium and the Zeroth Law of Thermodynamics The zeroth law of thermodynamics: If object A is in thermal equilibrium with object B, and object C is also in thermal equilibrium with object B, then objects A and C will be in thermal equilibrium if brought into thermal contact. That is, temperature is the only factor that determines whether two objects in thermal contact are in thermal equilibrium or not.
4 The Celsius scale: 172 Temperature Scales Water freezes at 0 Celsius. Water boils at 100 Celsius. The Fahrenheit scale: Water freezes at 32 Fahrenheit. Water boils at 212 Fahrenheit. Converting from Celsius to Fahrenheit: T ( F) 9 5 T C) 32 Converting from Fahrenheit to Celsius: T ( C) 5 9 T ( ( F) 32
5 172 Temperature Scales The pressure in a gas is proportional to its temperature. The proportionality constant is different for different gases, but they all reach zero pressure at the same temperature, which we call absolute zero: Absolute zero forms the basis of a temperature scale known as Absolute Scale or Kelvin Scale. The Kelvin scale is similar to the Celsius scale, except that the Kelvin scale has its zero at absolute zero. Conversion between a Celsius temperature and a Kelvin temperature: T(K) T( C)
6 172 Temperature Scales The three temperature scales compared:
7 174 Thermal Expansion Most substances expand when heated and contract when cooled. However the amount of expansion and contraction depends on the materials. Most solids generally expand in Length, Area and Volume as temperature increases. This can be understood as an increase in the amplitude of vibration of the atoms or molecules about their positions. Linear Expansion Experiments show that the change in length, ( L) is directly proportional to the change in temperature ( T) and also proportional to the original length (L 0 ) of the object. i.e L T, and L L 0 We can write the proportionality as an equation: The proportionality constant is called the coefficient of linear expansion.
8 174 Thermal Expansion We can write: Final length = Original length + Change in length: L L L L L 0 0 L 0 T L (1 0 T) Some typical coefficients of thermal expansion:
9 Area and Volume Expansion: The expansion of Area and Volume of a flat substance is derived from the linear expansion in two and three dimensions respectively: A 0 0 Definition of Coefficient of Volume Expansion, : 174 Thermal Expansion 2 A T A T Where: = 2 : Coefficient of area expansion 3 V0 T V T Where: = 3 : Coefficient of volume expansion V 0 V 3 V0 T V0 SI unit for : K 1 = ( C) 1 For liquids and gases, only the coefficient of volume expansion is defined: The Table shows some typical coefficients of volume expansion: T
10 174 Thermal Expansion Exercise 1: A steel railway track has a length of m when the temperature is 0 C. What is the length on a hot Melbourne day when the temperature is 40 C? Exercise 2: The steel bed of a suspension bridge is 200 m long at 20 C. If the extremes of the temperature to which it might be exposed are 30 C to 40 C. What total range of change in length must the expansion joints accommodate? (i.e. How much will it contract and expand?)
11 174 Thermal Expansion Exercise 3: A 70 L steel gas tank of a car is filled to the top with gasoline at 20 C. The car sits in the sun and the tank reaches a temperature of 40 C. How much gasoline do you expect to overflow from the tank? Exercise 4: A copper ball with a radius of 1.6 cm is heated from an initial temperature of 22 C to a final temperature of 680 C. Find the change in the volume of the ball and the final radius of the ball.
12 176 The Ideal Gas Laws and Absolute Temperature Behaviour of gases depends on the following properties of the gases: Pressure Volume Temperature Mass Number of Molecules Gases are the easiest state of matter to describe, as all ideal gases exhibit similar behavior. An ideal gas is one that is thin (dilute) enough, and far away enough from condensing, that the interactions between molecules can be ignored. Real Gases: The behavior of real gases is generally quite well approximated by that of an Ideal gas at low pressure (or low density), and at room temperature (or when T is not close to Liquefaction point).
13 177 The Ideal Gas Law We can describe the way the Pressure, P, of an ideal gas depends on: Temperature, T Number of molecules, N, and the Volume, V, from a few simple observations: (i) If the volume of an ideal gas is held constant, (as in the constant volume gas thermometer), we find that the pressure varies linearly with absolute temperature: (P T )
14 177 The Ideal Gas Law (ii) If the volume and temperature of a gas are kept constant, but more gas is added (such as in inflating a tire or basketball), the pressure will increase: (i.e. P N ) (iii) Finally, if the temperature and the number of molecules are held constant and the volume decreases, (such as sitting on a ball), the pressure increases. That is the pressure varies inversely with volume: (P 1/V or PV = constant)
15 177 The Ideal Gas Law Combining all three observations, we can write a mathematical expression fro the Pressure of a gas: where k is called the Boltzmann constant: Rearranging gives us the equation of state for an ideal gas: Instead of counting molecules, we can count moles. A mole is the amount of substance that contains as many atoms or molecules as there are atoms in 12 g of carbon12.
16 177 The Ideal Gas Law Experimentally, the number of atoms or molecules in one mole is given by Avogadro s number: Therefore, n moles of a gas will contain N = nn A molecules. Substituting this into the ideal gas equation: PV NkT nn A kt nrt Avogadro s number and the Boltzmann constant can be combined to form the Universal Gas Constant, R, defined as: R N SI unit A : k (L ( J/(mol J/(mol K) atm) /(mol K) molecules/mol)(1.38 K) J/K)
17 178 Problem Solving with the Ideal Gas Law The ideal gas law is an extremely useful tool. We often refer to Standard Conditions or Standard Temperature and Pressure (STP) which means: T = 273 K (0 C) and P =1.00 atm = N/m 2 =101.3 kpa When using the ideal gas law: the Temperature,T, must be given in Kelvin (K) and the pressure, P, must always be the Absolute pressure, not gauge pressure. In many situations, it is not necessary to use the value of R at all. For example, many problems involve a change in pressure, temperature, and volume of a fixed amount of gas. In this case: PV PV T nrt nr constant Since n and R remain constant, we can let P 1, V 1 and T 1 denote the initial variables and P 2, V 2 and T 2 denote the variables after the change (final conditions), then we can calculate the unknown variable using: PV 1 T 1 1 P V 2 T 2 2
18 178 Problem Solving with the Ideal Gas Law Boyle s law: is consistent with the ideal gas law. For a fixed quantity of gas, the volume of the gas is inversely proportional to the absolute pressure at constant temperature. (V 1/P, or PV = constant) These curves of constant temperature are called isotherms. PV 1 1 P2 V2 Fixed number of molecules, N; Fixed temperature, T Charles s law: is also consistent with the ideal gas law. The volume of a fixed quantity of gas is directly proportional to the absolute temperature if the pressure is kept constant. (V T, or V/T = constant) V T 1 1 V T 2 2 Fixed number of molecules, N; Fixed pressure, P
19 178 Problem Solving with the Ideal Gas Law Exercise 4: Determine the volume of 1.00 mole of any gas, assuming it behaves like an ideal gas at STP. Exercise 5: A person s lungs can hold 6.0 L (1L = 103 m 3 ) of air at a body temperature of 310 K and atmospheric pressure of 101 kpa. Given that the air is 21% oxygen, find the number of oxygen molecules in the lungs.
20 178 Problem Solving with the Ideal Gas Law Exercise 6: How many moles of air are in an inflated basketball? Assume that the pressure in the ball is 171 kpa, the temperature is 293 K, and the diameter of the ball is 30.0 cm. Exercise 7: An automobile tyre is filled to a gauge pressure of 200kPa at 10 C. After a drive of 100km, the temperature within the tyre rises to 40 C. What is the new pressure within the tyre at this temperature?
21 191 Heat and Mechanical Work Experimental work has shown that heat is another form of energy. James Joule used a device similar to this one to measure the mechanical equivalent of heat: As the mass falls, it turns the paddles in the water, which results in increase in water temperature. Thus Joule was able to show that mechanical energy (P.E.) is converted to heat One kilocalorie (kcal) is defined as the amount of heat needed to raise the temperature of 1 kg of water by 1 C (i.e. from14.5 C to 15.5 C) Joule used his experiments to find the mechanical equivalent of heat: 1 kcal = kj
22 191 Heat and Mechanical Work In studies of Nutrition, A different calorie is used. (Calorie with a capital C): 1 C = 1 kcal Heat, (Q) is the energy transferred from one object to another because of temperature difference Exercise 8: Working off the extra calories A 74 kg man eats too much ice cream on the order of 305 C. How many stairs of height 20.0 cm must he climb to work of the ice cream?
23 The Heat Capacity of an object is the amount of heat added to it divided by its rise in temperature: Specific Heat (c) 193 Specific Heats Q is positive if ΔT is positive; that is, if heat is added to a system. Q is negative if ΔT is negative; that is, if heat is removed from a system. The quantity of heat required to change the temperature of a given material is proportional to the mass, m of the material and to the temperature change, T. The specific heat, c, of any substance is defined as the amount of heat required to increase the temperature of 1kg of the substance by 1 C It can be rearranged as: Q = mc T
24 Here are some specific heats of various materials: 193 Specific Heats
25 194 Calorimetry Problem Solving using conservation of energy An isolated system is a closed system in which no heat energy is exchanged across its boundaries with the surroundings We use conservation of energy to figure out the final equilibrium temperature when two substances at different temperature are mixed and allowed to come to equilibrium within an isolated system. That is different parts of the system are at different temperatures. Heat flow from the part at higher temperature to the part at lower temperature within the system. Heat lost by one part of the system equals heat gained by the other part. i.e. Q = 0 Heat Lost = Heat Gained This is the basis for Calorimetry Technique: A calorimeter is a lightweight, insulated flask containing water. When an object is put in, it and the water come to thermal equilibrium. If the mass of the flask can be ignored, and the insulation prevents any heat exchange with the surroundings: 1. The final temperatures of the object and the water will be equal. 2. The total energy of the system is conserved. This allows us to calculate the specific heat of the object.
26 195 Latent Heats When two phases coexist, the temperature remains constant even if a small amount of heat is added. Instead of raising the temperature, the heat goes into changing the phase of the material melting ice, for example. i.e. certain amount of energy is used in this change of phase Figure shows temperature as a function of heat added to bring 1.0 kg of ice at 20 C to steam above 100 C
27 195 Latent Heats The heat required to convert from one phase to another is called the latent heat. The latent heat, L, is the heat that must be added to or removed from one kilogram of a substance to convert it from one phase to another. During the conversion process, the temperature of the system remains constant. Heat involved in a change of phase depends on the Latent Heat and also on the total mass of the substance. That is: The latent heat of fusion, (L F ), is the heat (required/released) to change from (solid to liquid/liquid to solid) phase Latent heat of vaporization, (L V ), is the heat (required/released) to change from (liquid to gas/gas to liquid) phase.
28 195 Latent Heats Table shows latent heats of fusion and vaporisation for various substances
29 195 Problem Solving using conservation of energy Exercise 9: The Cup Cools the Tea If 200 cm 3 of tea at 95 C is poured into a 150 g glass cup at 25 C. What will be the common final temperature, T, of the tea and cup at equilibrium assuming no heat flows to the surrounding? Exercise 10: Unknown Specific Heat determined by Calorimetry An Engineer wishes to determine the specific heat of a new metal alloy. A kg sample of the alloy is heated to 540 C. It is then quickly placed in kg of water at 10 C contained in a kg aluminum calorimeter cup. The final temperature of the system is 30.5 C. Calculate the specific heat of the alloy.
30 195 Problem Solving using conservation of energy Exercise 11: Will all Ice Melt Determine the final equilibrium temperature and phase (state) of the final mixture when 10 g of steam at 100 C is added to 80 g of ice 20 C.
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